HabbaDabbaDoo - Contract law does not apply to school admissions. It is entirely irrelevant. The applicable law is the Admissions Code and the School Standards and Framework Act 1998.
The case on which you attempt to rely was about admission appeals being conducted incorrectly, which straight away means it is not directly comparable. The school concerned had a pass mark. Pupils scoring below this mark would not be admitted even if the school had a place available. That is immediately different from this case. This school does not have a pass mark. It simply uses the test scores to place candidates in order. If it has a place available it must admit the highest scoring candidate regardless of their actual score. This is a crucial difference which means you cannot extrapolate from the LGO decision to this case.
I don't know what you think the OP should have done when offered this place. She contacted the LA several times and was assured the offer was correct. If she had rejected the offer on the grounds that she was convinced it was incorrect her daughter would have been left without a place at any school. The place at her third choice school would not have magically reappeared. Indeed, the LA would have been under no obligation to come up with a fresh offer at all.
Similar cases have been dealt with in the past by both the LGO and judicial review. Both the LGO and the courts have been clear that an offer made in error must be withdrawn within days or it stands. The courts agreed that an offer made in error could be validly withdrawn the same day. The LGO decided that an offer made in error must stand if not withdrawn within 3 days. Those are the rules. It matters not that the child shouldn't have been offered the place - that was, after all, the situation in both the LGO and judicial review cases.
Why must the offer stand? Because the parents and potentially the child can be massively disadvantaged otherwise. They may have committed significant sums of money on child care, uniform purchase, etc. based on the offer they have accepted. No-one is going to compensate them for that loss if the school is allowed to withdraw the place. Also, if a place at the first choice school is offered that overrides any place that would have been offered at the lower choices. If the first choice school is allowed to withdraw the offer weeks later the pupil may be left with a place at a sink school miles from home as the schools they might have got into if the offer hadn't been made in error (or had been withdrawn quickly) will have gone to other people. That is the reasoning used by the LGO in establishing the precedent.
You may not like the way the law stands on this appeal but it is clear. If a place offered in error is withdrawn incorrectly (as has happened here) that is maladministration that has deprived a child of a place. The fact that the place would never have been offered if the admission arrangements had been correctly administered is irrelevant.
The appeal panel in this case is clearly wrong in stating that the admission arrangements were administered correctly. If they had stated that there was maladministration but PermaShattered's child had not been disadvantaged that would have been a defendable position, although it would be clearly wrong based on precedents. But to state that there was no maladministration flies in the face of the facts. If the EFA is working to the same standards as the LGO (as they should be) the decision of the appeal panel will be overturned.